gmatassistance wrote:Can you please explain the following step - I'd like to better understand the shortcut as I usually am tight on time
"factor out 2^(x-2), which results in 2^(x-2) * (2^2 - 1) , which equals 2^(x-2)*3 "
Appreciate your help!
When we factor out, we're using the multiplication rule of exponents:
x^a * x^b = x^(a+b)
We need to factor out because you can't add or subtract terms with variables unless they have the same exponent.
For example, if we have:
2(x^2) + 3(x^3),
there's no simple way to combine those terms. However, if we have:
2(x^2) + 3(x^2)
we simply add the coefficients and get:
5(x^2).
So, if we want to add or subtract terms with variables, we need to equalize the exponents.
Before getting to this question, let's look at a simpler example:
2^8 - 2^6 = ?
We think to ourselves: we need to equalize the exponents. As a general rule, we always simplify to the smaller of the powers.
So, how can we express 2^8 with an exponent 6?
Well, using the multiplication rule noted up top, we can say that:
2^8 = 2^2 * 2^6 = 4 * 2^6
Plugging back into the question:
2^8 - 2^6 = 4(2^6) - 1(2^6) = 3(2^6)
(I rewrote 2^6 as 1(2^6) just to clarity the coefficient of that term.)
Now back to the question in this thread:
2^x - 2^(x-2)
(x-2) is the smaller exponent, so let's make that our main exponent for the expression.
So, we need to rewrite 2^x with a power of (x-2) instead of just x.
Well, using the multiplication rule of exponents:
2^x = 2^(x-2+2) = 2^(x-2) * 2^2 = 4(2^(x-2))
Substituting into the original expression:
2^x - 2^(x-2) = 4(2^(x-2)) - 1(2^(x-2)) = 3(2^(x-2))
